#500138

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,15)(16,26,28,27)(17,31,36,32)(18,25,34,22)(20,38,23,29)(21,40,24,35)(30,43,37,41)(33,42,44,39);(6,9)(10,12)(13,15,19,14)(16,27,28,26)(17,32,36,31)(18,22,34,25)(20,29,23,38)(21,35,24,40)(30,41,37,43)(33,39,44,42);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,33)(18,37)(19,28)(21,41)(22,42)(24,43)(25,39)(26,31)(27,32)(29,35)(30,34)(36,44)(38,40);(6,9)(10,12)(16,28)(20,23)(26,27)(29,38)(30,37)(33,44)(39,42)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407334);